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```848                              Appendix II.

2nd moment areas 02 a'2 are obtained on the same sides of the
vertical. Now the real value of I will be found by doubling our
results, for we have only used half of the section; hence,

I - 2(02 + a'2)j;2         and   Z - I = z^ + a'^y

or generally,

Moment of Resistance = /(2nd moment area)jy

if the reference distancey has been adopted. In cast-iron beams
jVt is always taken, which is less than yc (see p. 435), and the
resulting curves are slightly changed, but the construction ex-
plained must still be rigidly followed. It will be seen that this
method is superior to that at p. 430, though the latter is still left
in the text as sometimes convenient. (Seep. 1079.)

P. 441. Fixed Beams.—When beams have their ends
fixed as in Figs. 399 and 400, the Bm curve may be found
graphically, by supposing it the algebraic sum of two moment
curves, one caused by the ' action' of the free load, and the other
by the ' reaction' (a couple) in the wall itself. These opposing
curves must cover exactly equal areas: for, considering the
upper fibres of the beam say, the total effect of the load is to
shorten them; but, remembering that the total length of the beam
is unalterable on account of its fixedness, the reactionary couple
must entirely eliminate "the aforesaid compression. The sum of
the strains is therefore zero, and because Bm cc / oc b in uniform
sectioned beams, the sum of the Bm (average Bm x /) due to free
load must equal the sum of Bm due to wall couple.

Taking the special cases of Figs. 399 and 400, the first is
easily solved, but the second or uniformly loaded beam will be
further explained by Fig. 811. Now the Bm curve for a free
beam is a parabola, where b d, ~ W7-f-8; and the wall moment
is efgh) where eg = ^(bd\ to make areas A and B equal. The
final !Bm curve is shewn shaded at c. Lastly, to find contra-
flexure points, take any distance x between k and /.

t>     *         W/    ,.      W7   /W        *   r   x

Bm at x =----D = — - I— x.- -W • -

12              r»2    \2         /        2```